Question

Two identical firms (identical cost functions) operate on a market. For each of the following market demand curves and cost curves determine the Bertrand, Cournot, and Stackelberg outcomes (prices, quantities, and profits - for each firm, and at the market level). Also determine the collusive outcome (assuming the two firms form a cartel). Compare the outcomes. a) P = 200 − 2Q, T C = 50 + 10Q (PB = 10, PC = 73.33, PS = 57.5, PM = 105) b) P = 250 − Q, T C = 50Q (PB = 50, PC = 350/3, PS = 100, PM = 150) c) P = 1200 − Q, T C = 25 + 40Q (PB = 40, PC = 426.67, PS = 330, PM = 620) d) P = 160 − 2Q, T C = Q2(PB = 160/3, PC = 80, PS = 1600/21, PM = 96)

Answer 1

a)

P= 200-Q

for perfect competition P = MC

TC= 50+10Q, MC= 10

200-Q=10 ,Q= 190 substituting in original equation

P= 200-(190)

PB= 10

Profit = TR-TC= P * Q-TC = (10 * 190) - 50+10(190)= -50 loss

Now monopoly MR = MC

TR= 50Q-Q² and MR= 50-2Q and MC= 10

PS = 57.5

200-2Q = 10 ,Q= 95 substitute this in P=200-Q

P=200 - (95)

PM = 105

Profit= 105 * 95 - 50 + 10(20)= 3350

PC = 73.333

b)

P=250-Q ,TC= 50Q,

MC = 50

perfect= P=MC, 250-Q= 50,Q=200

PC= 250-200=50

profit = 50 X 200- 50(200) = 0

monopoly MR=MC

TR= 250Q-Q² ,MR=250-2Q and MC= 50

250-2Q= 50 ,Q=100

PS = 100

PM=250-100=150

profit = 150 * 100 - 50 * 100=10000

PC = (10000/150)+50 = 116.66

c)

P = 1200 − Q, T C = 25 + 40Q

MC = 40

1200-Q=40 ,Q= 1160 substituting in original equation

P= 1200-(1160) = 40,

PB= 40

Profit = TR-TC= P*Q-TC = (40X1160)- 1200+40(1160)= -1200 loss

Now monopoly MR=MC

TR= 1200Q-Q² and MR= 1200-2Q and MC= 40

1200-2Q=40 ,Q= 580 substitute this in P=1200-Q

P=1200- (580)= 620

PM = 620

Profit= 620 * 580 - 1200 + 40(580) = 5555200.

PS = 330

d)

P = 160 - 2Q, TC = 2Q, MC = 53.33

perfect= P=MC, 160 - 2Q= 53.33, Q= 53.33

P= 160-2(53.333)= 53.33

profit = 53.33*106.667- 53(1106.667)=0

monopoly MR=MC

TR= 160Q-Q² ,MR=160-2Q and MC= 53.33

PC = 80

PM=160-64=96

profit = 96 X 53.33 - 80*53.33 = 853.28

PS = 76.19